Distance to a Cycle in Undirected Graph

To find the minimum distance between each node and any node in the cycle, we can use a BFS (Breadth-First Search) approach. First, we’ll need to identify the cycle nodes. Then we’ll run a BFS from the cycle nodes and compute the distances for all nodes.

Here is the code that implements this approach:

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from collections import deque, defaultdict

class Solution:
    def distanceToCycle(self, n: int, edges: List[List[int]]) -> List[int]:
        graph = defaultdict(list)
        degrees = [0] * n

        # Create the graph and compute degrees of each node
        for u, v in edges:
            graph[u].append(v)
            graph[v].append(u)
            degrees[u] += 1
            degrees[v] += 1

        # Queue to process nodes
        queue = deque()

        # Add leaves (degree 1 nodes) to the queue
        for i in range(n):
            if degrees[i] == 1:
                queue.append(i)

        # Remove leaves until the cycle is left
        while len(queue) > 0:
            u = queue.popleft()
            degrees[u] = 0 # Marking it as processed
            for v in graph[u]:
                if degrees[v] > 1:
                    degrees[v] -= 1
                    if degrees[v] == 1:
                        queue.append(v)

        # Add remaining cycle nodes to the queue
        for i in range(n):
            if degrees[i] > 0:
                queue.append(i)

        # BFS to calculate distances to the cycle
        distances = [-1] * n
        level = 0
        while queue:
            size = len(queue)
            for i in range(size):
                node = queue.popleft()
                if distances[node] == -1: # If not visited
                    distances[node] = level
                    for neighbor in graph[node]:
                        queue.append(neighbor)
            level += 1

        return distances

The code constructs a graph and calculates the degrees of nodes. Then it iteratively removes leaf nodes (nodes with degree 1) until only the cycle nodes remain. Finally, it runs BFS from these cycle nodes and computes the distances for all nodes.

The time complexity of this solution is (O(n)), and the space complexity is also (O(n)), where (n) is the number of nodes in the graph.

Can you help me with finding the isomorphism for the LeetCode ? Please provide a concise response and if an exact mapping is not possible, specify it as an approximate mapping. If the titles of the problems are the same, select a different problem. Also, explain the reasoning behind your selection. Don’t mention anything about your training cut off date. Provide the best possible answer. Do not repeat the same point. Mention which problem is simpler. If there are multiple isomorphic problem. Arrange them from simple to more complex to your best possible knowledge. Do not add Leetcode to the problem name. The response format is:

The problem can be mapped to

10 Prerequisite LeetCode Problems

This involves graph theory, breadth-first search, and cycle detection in a graph. Here are 10 problems to prepare for it:

Number of Islands: This problem will help you get comfortable with the basics of graph traversal.
Course Schedule: This problem helps to understand the concept of cycle detection in a directed graph.
Flood Fill: It’s a good problem to get comfortable with concepts of Depth-First Search.
Graph Valid Tree: This problem asks whether a graph is a valid tree (i.e., it doesn’t contain any cycles).
Breadth First Search: It’s a classic BFS problem.
Find the Town Judge: This problem will help you understand some concepts of graph theory.
Breadth-first Search II: This problem will help you understand some additional concepts of breadth-first search.
Clone Graph: This problem will help you understand the concepts of graph copying and traversal.
Minimum Depth of Binary Tree: This problem is another classic breadth-first search problem.
Course Schedule II: This problem extends the concept of cycle detection and provides practice with topological sort.

Identifying Problem Isomorphism

Can you help me with finding the isomorphism for this problem?

Which problem does this problem map to the corresponding isomorphic problem on Leetcode ?

How did you identify that this problem is a variation of problem?

Problem Classification

Problem Statement:You are given a positive integer n representing the number of nodes in a connected undirected graph containing exactly one cycle. The nodes are numbered from 0 to n - 1 (inclusive).

You are also given a 2D integer array edges, where edges[i] = [node1i, node2i] denotes that there is a bidirectional edge connecting node1i and node2i in the graph.

The distance between two nodes a and b is defined to be the minimum number of edges that are needed to go from a to b.

Return an integer array answer of size n, where answer[i] is the minimum distance between the ith node and any node in the cycle.

Example 1:

Input: n = 7, edges = [[1,2],[2,4],[4,3],[3,1],[0,1],[5,2],[6,5]] Output: [1,0,0,0,0,1,2] Explanation: The nodes 1, 2, 3, and 4 form the cycle. The distance from 0 to 1 is 1. The distance from 1 to 1 is 0. The distance from 2 to 2 is 0. The distance from 3 to 3 is 0. The distance from 4 to 4 is 0. The distance from 5 to 2 is 1. The distance from 6 to 2 is 2. Example 2:

Input: n = 9, edges = [[0,1],[1,2],[0,2],[2,6],[6,7],[6,8],[0,3],[3,4],[3,5]] Output: [0,0,0,1,2,2,1,2,2] Explanation: The nodes 0, 1, and 2 form the cycle. The distance from 0 to 0 is 0. The distance from 1 to 1 is 0. The distance from 2 to 2 is 0. The distance from 3 to 1 is 1. The distance from 4 to 1 is 2. The distance from 5 to 1 is 2. The distance from 6 to 2 is 1. The distance from 7 to 2 is 2. The distance from 8 to 2 is 2.

Constraints:

3 <= n <= 105 edges.length == n edges[i].length == 2 0 <= node1i, node2i <= n - 1 node1i != node2i The graph is connected. The graph has exactly one cycle. There is at most one edge between any pair of vertices.

Analyze the provided problem statement. Categorize it based on its domain, ignoring ‘How’ it might be solved. Identify and list out the ‘What’ components. Based on these, further classify the problem. Explain your categorizations.

Visual Model of the Problem

How to visualize the problem statement for this problem?

Problem Restatement

Could you start by paraphrasing the problem statement in your own words? Try to distill the problem into its essential elements and make sure to clarify the requirements and constraints. This exercise should aid in understanding the problem better and aligning our thought process before jumping into solving it.

Abstract Representation of the Problem

Could you help me formulate an abstract representation of this problem?

Given this problem, how can we describe it in an abstract way that emphasizes the structure and key elements, without the specific real-world details?

Terminology

Are there any specialized terms, jargon, or technical concepts that are crucial to understanding this problem or solution? Could you define them and explain their role within the context of this problem?

Problem Simplification and Explanation

Could you please break down this problem into simpler terms? What are the key concepts involved and how do they interact? Can you also provide a metaphor or analogy to help me understand the problem better?

Constraints

Given the problem statement and the constraints provided, identify specific characteristics or conditions that can be exploited to our advantage in finding an efficient solution. Look for patterns or specific numerical ranges that could be useful in manipulating or interpreting the data.

What are the key insights from analyzing the constraints?

Case Analysis

Could you please provide additional examples or test cases that cover a wider range of the input space, including edge and boundary conditions? In doing so, could you also analyze each example to highlight different aspects of the problem, key constraints and potential pitfalls, as well as the reasoning behind the expected output for each case? This should help in generating key insights about the problem and ensuring the solution is robust and handles all possible scenarios.

Identification of Applicable Theoretical Concepts

Can you identify any mathematical or algorithmic concepts or properties that can be applied to simplify the problem or make it more manageable? Think about the nature of the operations or manipulations required by the problem statement. Are there existing theories, metrics, or methodologies in mathematics, computer science, or related fields that can be applied to calculate, measure, or perform these operations more effectively or efficiently?

Problem Breakdown and Solution Methodology

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Inference of Problem-Solving Approach from the Problem Statement

How did you infer from the problem statement that this problem can be solved using ?

Could you please provide a stepwise refinement of our approach to solving this problem?
How can we take the high-level solution approach and distill it into more granular, actionable steps?
Could you identify any parts of the problem that can be solved independently?
Are there any repeatable patterns within our solution?

Solution Approach and Analysis

Thought Process

Explain the thought process by thinking step by step to solve this problem from the problem statement and code the final solution. Write code in Python3. What are the cues in the problem statement? What direction does it suggest in the approach to the problem? Generate insights about the problem statement.

From Brute Force to Optimal Solution

Could you please begin by illustrating a brute force solution for this problem? After detailing and discussing the inefficiencies of the brute force approach, could you then guide us through the process of optimizing this solution? Please explain each step towards optimization, discussing the reasoning behind each decision made, and how it improves upon the previous solution. Also, could you show how these optimizations impact the time and space complexity of our solution?

Coding Constructs

Consider the following piece of complex software code.

What are the high-level problem-solving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a non-programmer, what would you say?
Can you identify the logical elements or constructs used in this code, independent of any programming language?
Could you describe the algorithmic approach used by this code in plain English?
What are the key steps or operations this code is performing on the input data, and why?
Can you identify the algorithmic patterns or strategies used by this code, irrespective of the specific programming language syntax?

Language Agnostic Coding Drills

Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.

Dissect the code and identify each distinct concept it contains. Remember, this process should be language-agnostic and generally applicable to most modern programming languages.
Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.
Next, describe the problem-solving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the step-by-step process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.

Targeted Drills in Python

Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Python-based coding drills for each of those concepts.

Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.
In addition to the general concepts, identify and write coding drills for any problem-specific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.
Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.

Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.